Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Show that any positive definite matrix A can be written as , where B is a positive definite matrix.

$A = B^{2}$ , where B is a positive definite matrix.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information:

$A = B^{2}$

Step 2: Determining is the B is positive definite Matrix:

Consider the quadratic form $q (\vec{x}) = \vec{x} \cdot A \vec{x}$ , where A is a $n \times n$ symmetric matrix. If $q (x)$ is positive for all nonzero $\vec{x}$ in $ℝ^{n}$ , we say A is positive definite. S is an orthogonal matrix that has the following properties:

$S^{- 1} A S = D$

When D is a diagonal matrix, all of the entries are positive. As a result, A can be written as:

$A = S D S^{- 1}$

We can write D as a diagonal matrix with positive diagonal elements.

$D = D_{1}^{2}$

Where is a diagonal matrix with positive diagonal entries, Equation can now be written as follows:

$A = {SDS}^{- 1}$

$\Rightarrow A = S (D_{1}^{2}) S^{- 1} \Rightarrow A = S (D_{1} \cdot D_{1}) S^{- 1} \Rightarrow A = (S D_{1} S^{- 1}) \cdot (S D_{1} S^{- 1}) \Rightarrow A = B^{2}$

${SBS}^{- 1} = D_{1}$ is a diagonal matrix with positive diagonal elements, hence B is positive definite.

Step 3: Determining the Result:

$A = B^{2}$ , where B is a positive definite matrix.

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Linear Algebra With Applications

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Short Answer

Show that any positive definite matrix A can be written as , where B is a positive definite matrix.

Step by Step Solution

Step 1: Given Information:

Step 2: Determining is the B is positive definite Matrix:

Step 3: Determining the Result:

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Linear Algebra With Applications

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Short Answer

Show that any positive definite matrix A can be written as , where B is a positive definite matrix.

Step by Step Solution

Step 1: Given Information:

Step 2: Determining is the B is positive definite Matrix:

Step 3: Determining the Result:

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